a) Overview of the Procedure in the Prior Art
In order to determine a THz spectrum, the photocurrent I(f) is measured which has oscillations on account of the measuring method used. The THz spectrum is given by the amplitude A(f) of this oscillating photocurrent.
Hitherto, the amplitude A(f) and thus the THz spectrum has been compiled from the absolute values of the maxima I(fmax) at the points f=fmax and from the absolute values of the minima I(fmin) at the points f=fmin.
As a result, the spectral resolution of the THz spectrum is limited to the oscillation period of the photocurrent measurement curve.
b) Measurement Set-Up for THz Spectra and Properties of the Photocurrent
Firstly, the measurement set-up for detecting the measurement signal [photocurrent I(f)] will be explained. Such a measurement set-up is also used for the method according to the invention, and the method according to the invention uses a measurement signal determined by such a measurement set-up.
FIG. 1 schematically illustrates a typical measurement set-up of a THz spectrometer. In order to generate the THz radiation, firstly two lasers 1, 2, which are operated at slightly different frequencies f1, f2, are mixed on a fiber-based 50/50 beam splitter 3. The wavelength of the lasers 1, 2 is unimportant a priori (it is usually approximately 1500 nm), but what is important is the frequency difference Δf=f1−f2 or Δω=ω1−ω2 with ω=2πf (hereinafter, reference is made primarily to the angular frequency ω). The frequency difference of the lasers 1, 2 is in the region of one terahertz, and can be varied by detuning the lasers 1, 2 with respect to one another. Upon the superimposition of the (coherent) laser beams 1a, 2a of the lasers 1, 2 at the beam splitter 3, cf. the first part 11 and the second part 12 of the superimposed laser radiation, a beat (laser beat) arises, wherein the envelope of the beat oscillates precisely with the frequency difference Δf of the two lasers 1, 2. This beat frequency is thus likewise in the terahertz range.
Behind the 50/50 beam splitter 3 the (now superimposed) laser beams are transmitted firstly (cf. the first part 11) to the emitter (THz transmitter) 13, where a THz radiation 14 is generated. Secondly (cf. the second part 12), the laser beams are transmitted to the detector (THz receiver) 15.
In the transmitter 13 (also referred to as Tx for short), as a result of the incident laser signal (that is to say the first part 11), charge carriers are excited to oscillate. However, on account of their mobility, said charge carriers can only follow the low frequency given by the envelope. Since the envelope oscillates at THz frequency, an electromagnetic wave is thus generated which has the angular frequency ωTHz=Δω (hereinafter, the subscript “THz” will be dispensed with in the case of the frequency ω). This THz wave then passes through a sample compartment 16 and is focused onto the receiver 15. In the beam path or in the sample compartment 16 (formed by a gas cell, for instance) it is possible to position a sample 17 (in particular a gaseous sample) having a thickness d whose spectral properties (transmission/absorption) are of interest; as a result, the THz radiation 14 becomes a characteristic transmission radiation 18.
The receiver 15 (also referred to as Rx for short) is a semiconductor-based detector which can be “armed to detection” as a result of the incidence of laser radiation. If it is impinged on by the laser beam arriving from the left in FIG. 1 (that is to say the second part 12), it is able to measure the field of the incident THz wave. In this case, the detected signal (that is to say the photocurrent I(f)) is proportional both to the electric field of the arriving laser beam ERxLaser(t, z) and to the electric field ERxTHz(t, z) of the THz wave at the location of the detector 15, that is to say that the photocurrent I(f) is proportional to the product of the two fields:I∝ERxTHz(t,z)·ERxLaser(t,z)  (1)
On the path to the detector 15, the two fields pick up different phases φ1, φ2 on their different paths from the photomixer (beam splitter 3) since they traverse different optical path lengths l1,l2. These two lengths l1,l2 are illustrated as dotted and solid curves in FIG. 2. The set-up is similar to an interferometer having two arms of different lengths, but in the present case the fields are multiplied by one another, rather than being added together as in the case of an interference experiment.
Both laser field and THz field can be written in a simplified way as a plane wave of frequency ω, wherein the laser traverses precisely the path length z=l2 and the THz wave traverses the path length z=l1:ERxTHz(t,z=l1)∝cos(ωt−φ1)  (2a)ERxLaser(t,z=l2)∝cos(ωt−φ2)  (2b)
The photocurrent then results as:I∝cos(ωt−φ1)·cos(ωt−φ2)  (3)
Using a cosine multiplication rule this expression can be simplified as:I∝cos(φ1−φ2)+cos(2ωt−φ1−φ2).  (4)
The second term oscillates with the frequency 2ω. This frequency is too fast to be able to be detected by the electronics; the term thus averages to zero. The first term is a DC component that can be measured. The phase difference φ1−φ2 can be expressed as:
                                          φ            1                    -                      φ            2                          =                                                            l                1                            -                              l                2                                      c                    ·                      ω            .                                              (        5        )            
That is to say that the phase difference is dependent on the THz frequency and the path length difference. In order to determine a spectrum S(ω) [or S(f)], the frequency ω is detuned and the path length difference in this case is left constant apart from the influence of the dispersion. The phase difference φ1−φ2 thus changes. The detected photocurrent therefore oscillates with the change in the phase difference.
                              I          ⁡                      (            ω            )                          ∝                  cos          ⁡                      (                                                                                l                    1                                    -                                      l                    2                                                  c                            ·              ω                        )                                              (        6        )            
The period of this oscillation is given by c/(l1−l2), that is to say that if the two path lengths are very similar, l1≈l2, then the oscillation of the photocurrent has a very long period as a function of the chosen terahertz frequency. Conversely, if the path lengths are very different, then the photocurrent oscillates very rapidly as a function of the terahertz frequency.
If a sample 17 is additionally introduced into the beam path, then equation (6) has to be modified. Introducing a sample 17 having the thickness d into the beam path may for example affect the amplitude of the radiation as a result of absorption of radiation, and affect the phase as a result of the refractive index n of the sample 17. Equation (6) therefore has to be supplemented by a frequency-dependent amplitude term A(ω) and an additional phase φ(ω) resulting from the dispersion of the sample 17. The photocurrent is obtained as a function of the THz frequency:
                              I          ⁡                      (            ω            )                          =                              A            ⁡                          (              ω              )                                ·                      cos            ⁡                          (                                                                                                                  l                        1                                            -                                              l                        2                                                              c                                    ·                  ω                                -                                  φ                  ⁡                                      (                    ω                    )                                                              )                                                          (        7        )            
For this phase term it holds true that:
                              φ          ⁡                      (            ω            )                          =                                                            nd                c                            ·              ω                        ⁢                                                  ⁢            with            ⁢                                                  ⁢            n                    =                      n            ⁡                          (              ω              )                                                          (        8        )            
A change in the oscillation period of the photocurrent I(ω) follows from the dispersion. The optically denser medium of the sample 17 lengthens one of the two partials beams 11, 12 (here partial beam 11) in the set-up.
The photocurrent I(f) during a typical measurement with a gas cell is illustrated in FIG. 3. The absorption of the terahertz radiation during passage through a gas cell was examined during the measurement. The spectral feature (“dip”) in the right-hand region of the curve (at approximately 1160 GHz), is important; it arises due to an absorption line of water. The line is spectrally narrowband since a low gas pressure was used in the gas cell.
The following section shows that this spectral feature is not adequately resolved by the prior art.
c) Determining the Amplitude According to the Prior Art
The method of data evaluation according to the prior art will now be described, and the limitation that arises as a result will be explained.
The functional dependence of the photocurrent on the THz frequency ω is given by equation (7). This equation contains the amplitude A(ω) and the phase φ(ω) as frequency-dependent variables. Both variables have to be determined from the measurement curve, only the amplitude A(ω) being relevant for the determination of an absorption spectrum (of the “absorbance”). The determination of two unknowns from only one measurement curve is a non-trivial, inverse problem since at every point of the curve it is necessary to determine two variables from one equation.
Hitherto (cf. reference [2]), the amplitude A(ω) was determined at the maxima ω=ωmax and minima ω=ωmin, of the curve I(ω) since it is known at these points that the argument of the cosine is the even and respectively odd multiple of π:
                                                                                                              l                    1                                    -                                      l                    2                                                  c                            ·                              ω                max                                      -                          φ              ⁡                              (                                  ω                  max                                )                                              =                                    n              ·              2                        ⁢            π                          ,                                  ⁢                  n          =          0                ,        1        ,        2        ,                  …          ⁢                                          ⁢          for          ⁢                                          ⁢          maxima          ⁢                                          ⁢          and                                    (                  9          ⁢          a                )                                                                                                                          l                    1                                    -                                      l                    2                                                  c                            ·                              ω                min                                      -                          φ              ⁡                              (                                  ω                  min                                )                                              =                                    (                                                2                  ⁢                  m                                +                1                            )                        ·            π                          ,                                  ⁢                  m          =          0                ,        1        ,        2        ,                  …          ⁢                                          ⁢          for          ⁢                                          ⁢                      minima            .                                              (                  9          ⁢          b                )            
Thus, at these points the determination of the phase φ(ω) is obviated (since the cosine results here in each case as 1 or as −1) and the determination of A(ω) is unambiguous. A corresponding calculation can be integrated into the measurement software, and the result A(ωmin/max) thereof is also referred to as the “envelope”. FIG. 4 shows an amplitude curve determined in such a way according to the prior art (circles linked by dashed line). It is evident that the amplitude curve calculated in this way contains only few points, and the above-described dip in the right-hand part of the curve of the photocurrent I(f) is not resolved in the envelope.
If the absorption dip were wider than the distance between multiple extrema, then the limited resolution would not be a problem. Such an example is illustrated in FIG. 5. In the center of the spectrum it is observed that the photocurrent dips over a width of approximately 5 GHz that is to say that the oscillations have a significantly smaller amplitude. In this case, too, an absorption line of water vapor is involved, but this line was measured at high pressure. As a result of the high pressure, so-called collisional broadening occurs and the absorption line has a significantly higher linewidth than in FIG. 3 and FIG. 4. The absorption line in FIG. 5 extends over approximately oscillations in the photocurrent and can thus be resolved by the conventional determination of the envelope (amplitude curve, circles linked by dashed line).
d) On the Definition of the Term “Resolution” in the Evaluation of the Photocurrent Curve
Hereinafter, repeated use is made of the term “resolution” in comparative form (“higher resolution” and “lower resolution”). It is therefore expedient to define the term in the context of the determination of the amplitude A(f) of the photocurrent I(f).
Let there be a periodically oscillating photocurrent in accordance with eq. (7). This photocurrent curve is measured for discrete frequency values fi (with i: counting index). We assume that within a period p=c/(l1−l2) of the photocurrent curve there are m measurement values with the spacing δf=|fi+1−fi|. With the evaluation according to the prior art, two points for the amplitude curve are obtained within a period—one point for the maximum within the period and one point for the minimum. Hereinafter, these points are called “result points” r. The spectral resolution capability Δf using the method in accordance with the prior art (with r=2) is thus
            Δ      ⁢                          ⁢      f        =                  p        r            =                                    m            ·            δ                    ⁢                                          ⁢          f                2              ,and is related directly to the oscillation period p (also see the following section). In order that the oscillation can be identified as such, there should be a sufficient number of data points between a minimum and a neighboring maximum. The more data points there are therebetween, the more accurately the extremum points can be identified, that is to say that for the spacing of the measurement values δf it holds true that: δf<<p=c/(l1−l2). As a consequence, it is necessary to measure the photocurrent curve in steps of δf, but the evaluation method according to the prior art yields only a resolution Δf, that is coarser by the factor m/2. If it were desired to improve the resolution capability, then firstly the oscillation period p must be reduced (also see the following section). At the same time, however, the step size δf must also be decreased in order to correctly detect the oscillation. If the intention is to pass through the same spectral range as before, then the total number of measured points of the measurement increases. A fixed integration time is required for each measurement point; the time requirement of a measurement thus increases.
As will be shown later, in the case of the present invention the resolution is limited only by the step size δf; therefore, for the same time requirement, the invention offers a resolution improved by the factor m/2.
e) Relationship Between Evaluation Method and Physical Set-Up of the Measurement
As represented above in equations (6) and (7), the oscillation period of the photocurrent is proportional to the path length difference l1−l2. In the concrete experiment, this path length difference is set by the length of the optical fibers between beam splitter and transmitter and receiver, respectively. In a typical experimental set-up, by way of example, the path length difference is l1−l2≈21 cm, which leads to an oscillation period of
  p  =            c                        l          1                -                  l          2                      =          1.4      ⁢                          ⁢              GHz        .            
If the evaluation method described above is then used, and the extremum points are used in the calculation of the amplitude of the photocurrent, data points are obtained with a resolution of 0.7 GHz (the minima are also taken into account, that is to say that by seeking the maxima of the absolute value of the photocurrent, data points with the spacing from a maximum to the adjacent minimum are thus obtained). That means that the resolution is directly coupled to the “physical configuration of the set-up” (of the fiber length chosen). If there is a desire to change the possible resolution, then the configuration must be changed, and the evaluation method thus has a repercussion on the experiment. This repercussion is eliminated in the context of the invention.
f) Targeted Alteration of the Optical Path Length for Controlling the Phase φ(ω)
Reference [1] describes an instrumental technology which can be used as an alternative to the evaluation technique shown above and which makes it possible to determine the amplitude A(ω) for all the measurement points ω . In the case of this technology, a so-called fiber stretcher is incorporated into the optical fiber connecting the beam splitter to the THz transmitter or respectively to the receiver. With said fiber stretcher, the optical fiber is mechanically elongated and the optical path length l1 or respectively l2is thus altered in a targeted manner. In the case of a change in the path length l1, for instance, the argument of the cosine in equation (7) can be rewritten as
                                          l            1            ′                    -                      l            2                          c            ·      ω        -          φ      ⁡              (        ω        )              ,where l′1=l1+lfiber,wherein lfiber is the set additional length of the fiber. The length of the lfiber can be set in a targeted manner for each measurement point ω; lfiber thus becomes a function of the frequency lfiber(ω). It is instructive to consolidate the term lfiber(ω) with the frequency-dependent phase φ(ω):
                                          l            1                    -                      l            2                          c            ·      ω        +                                                                      l                fiber                            ⁡                              (                ω                )                                      c                    ⁢          ω                -                  φ          ⁡                      (            ω            )                                      ︸                  =                                    φ              ~                        ⁡                          (              ω              )                                            ,that is to say that the use of the fiber stretcher leads to a control of the phase (ω). By way of example, it is then possible to set (ω) for each frequency ω such that the argument of the cosine is always precisely an integer multiple of 2π and the cosine term in equation (7) always yields 1:
      I    ⁡          (      ω      )        =                    A        ⁡                  (          ω          )                    ·                        cos          ⁡                      (                                                                                                      l                      1                                        -                                          l                      2                                                        c                                ·                ω                            -                                                φ                  ~                                ⁡                                  (                  ω                  )                                                      )                                    ︸                      =            1                                =                  A        ⁡                  (          ω          )                    .      
The determination of the amplitude A(ω) is thus unambiguous for every measurement point and the limitation to the extremum points of the photocurrent curve is obviated.
Reference [1] explains that the alteration of the fiber length is applied periodically, that is to say that the phase (ω) is varied periodically between 0 and 2π. The precise procedure in [1] is not set out exactly; the authors write that the photocurrent is determined at the receiver by averaging the signal.
Although the method using the fiber stretcher is not subject to the limitations of the evaluation method described above, the method entails the following disadvantages:                an additional component (fiber stretcher) with a high cost factor has to be integrated into the set-up.        The phase (ω) has to be varied for each measurement point ω; the measurement time is lengthened as a result.        
With the method according to the invention, the use of a fiber stretcher is not necessary, as a result of which the experimental set-up is simplified, and a variation of the phase for each measurement point or each measurement frequency ω is not required, as a result of which the measurement time is shortened.